A classic result states that the equation x³＋px²＋qx＋r＝0 with real coefficients p, q, r has positive roots iff p＜0, q＞0, r＜0 and -27r² - 2p(2p² - 9q)r + q²(p² - 4q) ⩾ 0 (see for example this question). 
However, Maple appears unable to find the condition: 

a, b, c := allvalues(RootOf(x^3 + p*x^2 + q*x + r, x), 'implicit'):
RealDomain:-solve({a, b, c} >~ 0, [p, q, r]);
 = 
Warning, solutions may have been lost
                               []

Is there a way to get the above conditions in Maple with as little human intervention as possible (I mean, without a priori knowledge of the theory of polynomials)? 

Edit. An interesting problem is when these three positive roots can further be the lengths of sides of a triangle. For reference, here are some (unenlightening) results from some other software: 

I don't know how make my graph be beter for real part and imaginary part and abs part which part how work with parameter can any one explain on this example?

G.mw

I have this file
 

restart:
F := proc(ee,LL)
  uses InertForm, Typesetting;
  mrow(Typeset(Display(eval(eval(MakeInert(factor(ee)),[`%*`=`*`])
                                 =MakeInert(subs(b=MakeInert(b*y)/y,
                        a=MakeInert(a*x)/x,p)),[a,b]=~LL),
                       inert=false)),
       mo("="),Typeset(eval(ee,[a,b]=~LL)))
end proc:
p := (a*x)^2 - 2*a*x*b*y + (b*y)^2:
L := [[sqrt(2),3],[2,5],[3,12],[1/3,5/7]];
ans := F~(p, L):
print~(ans):

How can I put the results like this
 

\documentclass[12pt,a4paper]{article}
\usepackage[left=2cm, right=2cm, top=2cm, bottom=2cm]{geometry}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{enumitem}
\theoremstyle{definition}
\newtheorem{ex}{Exercise}
\begin{document}
\begin{ex}
\[(\sqrt{2} x-3 y)^2=(\sqrt{2} \cdot x)^2-2 \cdot(\sqrt{2} \cdot x) \cdot(3 \cdot y)+(3 \cdot y)^2=2 x^2-6 \sqrt{2} x y+9 y^2.\]
\end{ex}

\begin{ex}
\[(2 x-5 y)^2=(2 \cdot x)^2-2 \cdot(2 \cdot x) \cdot(5 \cdot y)+(5 \cdot y)^2=4 x^2-20 x y+25 y^2. \]
\end{ex}

\begin{ex}
\[(3 x-12 y)^2=(3 \cdot x)^2-2 \cdot(3 \cdot x) \cdot(12 \cdot y)+(12 \cdot y)^2=9 x^2-72 x y+144 y^2. \]
\end{ex}

\begin{ex}
\[\left(\frac{x}{3}-\frac{5 y}{7}\right)^2=\left(\frac{1}{3} \cdot x\right)^2-2 \cdot\left(\frac{1}{3} \cdot x\right) \cdot\left(\frac{5}{7} \cdot y\right)+\left(\frac{5}{7} \cdot y\right)^2=\frac{1}{9} x^2-\frac{10}{21} x y+\frac{25}{49} y^2. \]
\end{ex}
 
\end{document} 

I've recently started to move my Maple software development from pure Maple to Visual Studio Code with GitHub. Previous code attachments in workbooks are now moved to its own folders, and fetched by the $include instruction.

The approach works quite well, but I have experienced a strange effect. While the Maple workbook itself is in the local GitHub folder, I do experience that when documents within the Maple workbook, after a while the code is fetched from the backup location. This doesn't work, as the $included files can't be found there. Only way is to close the workbook, and open it again.

First image shows location of Maple workbook, second location of document (backup position, error), third location of document after reopening workbook (working).

For a month I did do not get email notfications.

I checked spam folders, settings and email filters.

Is it only me?

The attached sheet contains equations H1 to H6 and K1 to K3. What data do I need to modify to ensure that the values of H1 to H6 fall between 0 and 1, and K1 to K3 are negative? The parameter ranges are also given in the sheet. Is there a method to achieve this?

rouhg.mw

i did two case of this equation and odetest is worked good but in this case the odetest is not worked well anyone can determine what is mistake ?

F_P_Correct_case_three.mw

Dear Maple user I am facing error while running the codes  to plot the graph for two data sets .

I am attaching the files.

Error_in_Display_1.mw

Why does dsolve not call odetest by default before a solution is returned?

I mean, why do I have test each result separately. dsolve could have an odetest option (default=true).

In case of discrepancies dsolve could inform the user and suggest to call dsolve with odetest=false and run odetest separately to analyse the problem.

Set up this way, dsolve would never return potentially incorrect solutions that do not pass odetest.

HTR.mw

In above problem, Additionally How to  plot  heat transfer rate  Q versus L^2  for distinct porosity parmeters(Sh) , using  heat transfer rate formula, Q = (q*L)/(k*A*T[b])=theta'(1).

using  [Sh = 0.1, L^2 = 0.1, Nr =0 .1, Ha =0 .1, Pe = 0.1],  [Sh = 0.3, L^2 = 0.3, Nr = 0.1, Ha = 0.1, Pe =0 .1],   [Sh = 0.5, L^2 =0 .5, Nr =0 .1, Ha = 0.1, Pe =0 .1].

How can I find Mean, Median, Quartiles, Variance, StandardDeviation of data in this table

I use Mathamatica and get the result

Clear["Global`*"]
boundaries = Range[0, 10, 5/2];
frequencies = {18, 11, 13, 6};
binMeans = Mean /@ Partition[boundaries, 2, 1];
weighted = WeightedData[binMeans, frequencies];
weightedHist = HistogramDistribution[weighted, {5/2}];
Through[{Mean, Median, Quartiles, Variance, StandardDeviation}[
  weightedHist]]

I trying to simplify expressions for lines so no higher order terms. factor and op seperate out what I need but how do I select the one with the variables in this case x,y. I cant depent on this always been the last one returned from the op command.

Download 2024-09-11_Has_Select_Question.mw

I get my on results but the results are not the same please help me if i did any mistake in my code

symmetry_PDESYS_3_time_fraction[1].mw

This code is working for function f1 but not for f2
f2 := (x,y)->9*x^2-24*x*y+16*y^2+10*x-70*y + 175;
Why this code is not working for f2 ?
unprotect(D);
f1:= (x, y) -> 3*x^2 - 3*y*x + 6*y^2 - 6*x + 7*y - 9;
coeffs(f(x, y));
A, B, C, D, E, F := %;
theta := 1/2*arctan(B/(A - C));
solve({-2*A*xc - B*yc = D, -B*xc - 2*C*yc = E});
assign(%);
x := xcan*cos(theta) - ycan*sin(theta) + xc;
y := xcan*sin(theta) + ycan*cos(theta) + yc;
Eq := simplify(expand(f1(x, y)));
xcan^2/simplify(sqrt(-tcoeff(Eq)/coeff(Eq, xcan^2)))^`2` + ycan^2/simplify(sqrt(-tcoeff(Eq)/coeff(Eq, ycan^2)))^`2` = 1;

Thank you